3 Answers
3

a) is true. generically there is no finger print that identifies one electron from another, so, given a proton, any electron can occupy its energy levels and make it into hydrogen, because all electrons are identical.

When one talks of specific energy levels in which there is an electron, then the identity qualifies the whole wavefunction, potential and energy level and electron. The answer to b) is that no,states with electrons in different energy levels are not identical to each other, even though the specific electrons have no fingerprint to identify them.

so b) is answered by only in degenerate levels the identity of electrons comes to play.

The source of question seems to be a confusion of what it means for two electrons "to be on the same energy level" or "to be on a different energy level".

Let $ψ_1(x),ψ_2(x)$ be the eigenstates of a single electron with energies $E_1$ and $E_2$ respectively. The wave function for two electrons where both electrons are independent and in the different energy levels is:

$$ Ψ(x_1,x_2) = ψ_1(x_1)ψ_2(x_2) - ψ_2(x_1)ψ_1(x_2) .$$

As you can see, this wave function is antisymmetric under permutations: we cannot tell which electron is in which energy level; we have no means of distinguishing them. All electrons are identical.

Finally, a remark on occuptation numbers: they behave like any other operator. They have eigenvalues, but a generic quantum mechanical state is usually not an eigenstate. In other words, a quantum state is in a superposition of eigenstates of the occuptation number operator(s). The common picture electrons sitting at each energy level is fundamentally misleading, but it does make sense when doing thermodynamics when we essentially assign "occupation probabilities".

In atomic theory, the state of the system is described by the occupation of the orbitals, or states. Electrons have no identity, ie. it doesn't matter which one occupies which orbital. The best way to thing of this is that the state is not described like "electron 1 is in state X, electron 2 is in state Y, and so on" but instead "there are $n_x$ electrons in state X, $n_y$ electrons in state Y, and so on". Since electrons are fermions $n$ is always 0 or 1.

On the other hand, if you measure a radiation of a certain wavelength, and you see that this corresponds to the energy difference between two orbitals X and Y, you can say that "the" electron in state Y jumped down to state X.